Reddit mentions of Complex Analysis (Princeton Lectures in Analysis, No. 2)

I'm on vacation, which means it's self study time. Definitely my favorite way to learn math.

I know enough algebra and modern algebraic geometry at this point that the best way to learn more alg geo should be to learn a bunch of differential geometry, so that's what I'm doing. A friend and I have been working through Global Calculus by Ramanan, and I'm looking to do directed reading from it once the school year starts.

We started with Lee's Introduction to Smooth Manifolds but it's dreadfully boring, index heavy, covers too many topics without indicating which ones are important, and doesn't block off important definitions and remarks. Also I don't really like einstein notation. Global Calculus strikes a really nice balance in which it uses categorical concepts to streamline and simplify the core ideas but doesn't randomly categorify everything for no reason (eg nLab on most things). However it is really, really dense, with my friend and I having spent hours discussing half a page at times, and lightly peppered with mistakes and little obscurisms.

The same friend and I, and our grad student mentor, are also polishing up a jointly written paper for submission, which will be my first. It's in combinatorics, and while it's very grounded, concrete stuff, a very non-concrete category of CW complexes popped up while we were trying to extend our ideas. That reminded me to reread Emily Riehl's excellent expositional paper A Leisurely Introduction to Simplicial Sets, so that's what I'm doing.

I took a kind of mini-course in complex analysis that went through about the first three chapters in Stein and Shakarchi's Complex Analysis. I might be taking a Reimann surfaces course next quarter, so I've been working through the stuff I didn't see to make sure I'm up to speed with what somebody taking a standard intro course would know. I got to cut out most of the book because I have no interest in number theory and because no intro course is going to assume prior knowledge of fourier analysis, so at this point probably the only thing I have left to do is pick out and do the interesting exercises related to the Reimann mapping theorem. I say probably because I'm not sure whether I want to go through the material about conformal mappings into polygons. I have book format version of A Concise Course in Complex Analysis by Schlag arriving soon, so once I'm done with Stein that'll probably be my secondary work source after burning out on diff geo each day.

I have grad school applications this year, so I'm also working on getting my shit together for that. I have the math GRE in october and I barely remember a single trick for computing integrals so wish me luck.

Stein and Shakarchi: http://www.amazon.com/Complex-Analysis-Princeton-Lectures-No/dp/0691113858

Fair enough! That makes sense.

Since you did well in the topic I'd assume you know of the basics pretty well. If you'd still like to brush up on the topics, I really like Ahlfors' text. Its not everybody's cup of tea and its a bit terse but for someone looking for a second look at Complex Analysis it should be doable. If not then go for something less dense like Stein/Sarkachi or Gamelin.

If you are looking for topics then allow me to suggest you one: if you liked Geometry in university then I highly recommend looking into Complex Geometry, which is the study of complex manifolds. Holomorphic functions (or complex analytic, depending on what text you used) in [; \mathbb{C} ;] have really interesting/wacky properties as is (think analytic continuation, Louisville etc.). Now imagine the fun when you lift that up to manifolds! There are lots of tie ins with algebraic geometry as well(more so, imo than with differential geometry) so if that's something you liked, it is worth looking into.

I have to admit I don't know as much about this topic as I should but I think Complex Geometry is quite cool and if you found geometry in university at all interesting, I think it will be a fulfilling topic for you. Let me know if that sounds at all cool then we can talk literature.

This is a good option: http://www.amazon.com/Complex-Analysis-Princeton-Lectures/dp/0691113858

At what level? I'm really enjoying Stein & Shakarchi but that's closer to graduate level.